Commutative ring/Ideal/Definition

A subset ${\displaystyle {}{\mathfrak {a}}}$ of a commutative ring ${\displaystyle {}R}$ is called an ideal, if the following conditions are fulfilled:

1. ${\displaystyle {}0\in {\mathfrak {a}}}$.
2. For all ${\displaystyle {}a,b\in {\mathfrak {a}}}$, we have ${\displaystyle {}a+b\in {\mathfrak {a}}}$.
3. For all ${\displaystyle {}a\in {\mathfrak {a}}}$ and ${\displaystyle {}r\in R}$, we have ${\displaystyle {}ra\in {\mathfrak {a}}}$.